Rings whose cozero-divisor graph has crosscap number at most two

Abstract

Let [Formula: see text] be a commutative ring with identity. The cozero-divisor graph of [Formula: see text], denoted by [Formula: see text], is a graph whose vertex set is [Formula: see text], the set of all non-zero and non-unit elements of [Formula: see text]. Two distinct vertices [Formula: see text] and [Formula: see text] in [Formula: see text] are adjacent if and only if [Formula: see text] and [Formula: see text]. The Crosscap of a graph [Formula: see text], denoted by [Formula: see text], is the minimum integer [Formula: see text] such that the graph can be embedded in the non-orientable surface [Formula: see text]. The planar graph is called [Formula: see text]-outerplanar if removing all the vertices incident on the outer face yields a [Formula: see text]-outerplanar. The Outerplanarity index of a graph [Formula: see text] is the smallest [Formula: see text] such that [Formula: see text] is [Formula: see text]-outerplanar. In this paper, we characterize the class of rings [Formula: see text] (up to isomorphism) for which [Formula: see text]. Further we characterize all finite rings [Formula: see text] (up to isomorphism) for which [Formula: see text] has an outerplanarity index two.